Theorem oa4gto4u 956

Description: A "universal" 4-OA derived from the Godowski/Greechie form. The hypotheses are the Godowski/Greechie form of the proper 4-OA and substitutions into it.

Hypotheses

Ref	Expression
oa4gto4u.1	((e ->1 d) ^ (((e ^ f) v ((e ->1 d) ^ (f ->1 d))) v (((e ^ g) v ((e ->1 d) ^ (g ->1 d))) ^ ((f ^ g) v ((f ->1 d) ^ (g ->1 d)))))) =< (f ->1 d)
oa4gto4u.2	f = (a ->1 d)
oa4gto4u3	e = (b ->1 d)
oa4gto4u.4	g = (c ->1 d)

Assertion

Ref	Expression
oa4gto4u	((a ->1 d) ^ ((a_|_ ->1 d) v ((b_|_ ->1 d) ^ ((((a ->1 d) ^ (b ->1 d)) v ((a_|_ ->1 d) ^ (b_|_ ->1 d))) v ((((a ->1 d) ^ (c ->1 d)) v ((a_|_ ->1 d) ^ (c_|_ ->1 d))) ^ (((b ->1 d) ^ (c ->1 d)) v ((b_|_ ->1 d) ^ (c_|_ ->1 d)))))))) =< d

Proof of Theorem oa4gto4u

Step	Hyp	Ref	Expression
1		oa4gto4u.2	. . . 4 f = (a ->1 d)
2	1	ud1lem0b 248	. . . . . 6 (f ->1 d) = ((a ->1 d) ->1 d)
3		u1lem12 763	. . . . . 6 ((a ->1 d) ->1 d) = (a_|_ ->1 d)
4	2, 3	ax-r2 35	. . . . 5 (f ->1 d) = (a_|_ ->1 d)
5		oa4gto4u3	. . . . . . . 8 e = (b ->1 d)
6	5	ud1lem0b 248	. . . . . . 7 (e ->1 d) = ((b ->1 d) ->1 d)
7		u1lem12 763	. . . . . . 7 ((b ->1 d) ->1 d) = (b_|_ ->1 d)
8	6, 7	ax-r2 35	. . . . . 6 (e ->1 d) = (b_|_ ->1 d)
9		ancom 68	. . . . . . . . 9 (e ^ f) = (f ^ e)
10	1, 5	2an 72	. . . . . . . . 9 (f ^ e) = ((a ->1 d) ^ (b ->1 d))
11	9, 10	ax-r2 35	. . . . . . . 8 (e ^ f) = ((a ->1 d) ^ (b ->1 d))
12		ancom 68	. . . . . . . . 9 ((e ->1 d) ^ (f ->1 d)) = ((f ->1 d) ^ (e ->1 d))
13	4, 8	2an 72	. . . . . . . . 9 ((f ->1 d) ^ (e ->1 d)) = ((a_|_ ->1 d) ^ (b_|_ ->1 d))
14	12, 13	ax-r2 35	. . . . . . . 8 ((e ->1 d) ^ (f ->1 d)) = ((a_|_ ->1 d) ^ (b_|_ ->1 d))
15	11, 14	2or 67	. . . . . . 7 ((e ^ f) v ((e ->1 d) ^ (f ->1 d))) = (((a ->1 d) ^ (b ->1 d)) v ((a_|_ ->1 d) ^ (b_|_ ->1 d)))
16		ancom 68	. . . . . . . 8 (((e ^ g) v ((e ->1 d) ^ (g ->1 d))) ^ ((f ^ g) v ((f ->1 d) ^ (g ->1 d)))) = (((f ^ g) v ((f ->1 d) ^ (g ->1 d))) ^ ((e ^ g) v ((e ->1 d) ^ (g ->1 d))))
17		oa4gto4u.4	. . . . . . . . . . 11 g = (c ->1 d)
18	1, 17	2an 72	. . . . . . . . . 10 (f ^ g) = ((a ->1 d) ^ (c ->1 d))
19	17	ud1lem0b 248	. . . . . . . . . . . 12 (g ->1 d) = ((c ->1 d) ->1 d)
20		u1lem12 763	. . . . . . . . . . . 12 ((c ->1 d) ->1 d) = (c_|_ ->1 d)
21	19, 20	ax-r2 35	. . . . . . . . . . 11 (g ->1 d) = (c_|_ ->1 d)
22	4, 21	2an 72	. . . . . . . . . 10 ((f ->1 d) ^ (g ->1 d)) = ((a_|_ ->1 d) ^ (c_|_ ->1 d))
23	18, 22	2or 67	. . . . . . . . 9 ((f ^ g) v ((f ->1 d) ^ (g ->1 d))) = (((a ->1 d) ^ (c ->1 d)) v ((a_|_ ->1 d) ^ (c_|_ ->1 d)))
24	5, 17	2an 72	. . . . . . . . . 10 (e ^ g) = ((b ->1 d) ^ (c ->1 d))
25	8, 21	2an 72	. . . . . . . . . 10 ((e ->1 d) ^ (g ->1 d)) = ((b_|_ ->1 d) ^ (c_|_ ->1 d))
26	24, 25	2or 67	. . . . . . . . 9 ((e ^ g) v ((e ->1 d) ^ (g ->1 d))) = (((b ->1 d) ^ (c ->1 d)) v ((b_|_ ->1 d) ^ (c_|_ ->1 d)))
27	23, 26	2an 72	. . . . . . . 8 (((f ^ g) v ((f ->1 d) ^ (g ->1 d))) ^ ((e ^ g) v ((e ->1 d) ^ (g ->1 d)))) = ((((a ->1 d) ^ (c ->1 d)) v ((a_|_ ->1 d) ^ (c_|_ ->1 d))) ^ (((b ->1 d) ^ (c ->1 d)) v ((b_|_ ->1 d) ^ (c_|_ ->1 d))))
28	16, 27	ax-r2 35	. . . . . . 7 (((e ^ g) v ((e ->1 d) ^ (g ->1 d))) ^ ((f ^ g) v ((f ->1 d) ^ (g ->1 d)))) = ((((a ->1 d) ^ (c ->1 d)) v ((a_|_ ->1 d) ^ (c_|_ ->1 d))) ^ (((b ->1 d) ^ (c ->1 d)) v ((b_|_ ->1 d) ^ (c_|_ ->1 d))))
29	15, 28	2or 67	. . . . . 6 (((e ^ f) v ((e ->1 d) ^ (f ->1 d))) v (((e ^ g) v ((e ->1 d) ^ (g ->1 d))) ^ ((f ^ g) v ((f ->1 d) ^ (g ->1 d))))) = ((((a ->1 d) ^ (b ->1 d)) v ((a_|_ ->1 d) ^ (b_|_ ->1 d))) v ((((a ->1 d) ^ (c ->1 d)) v ((a_|_ ->1 d) ^ (c_|_ ->1 d))) ^ (((b ->1 d) ^ (c ->1 d)) v ((b_|_ ->1 d) ^ (c_|_ ->1 d)))))
30	8, 29	2an 72	. . . . 5 ((e ->1 d) ^ (((e ^ f) v ((e ->1 d) ^ (f ->1 d))) v (((e ^ g) v ((e ->1 d) ^ (g ->1 d))) ^ ((f ^ g) v ((f ->1 d) ^ (g ->1 d)))))) = ((b_|_ ->1 d) ^ ((((a ->1 d) ^ (b ->1 d)) v ((a_|_ ->1 d) ^ (b_|_ ->1 d))) v ((((a ->1 d) ^ (c ->1 d)) v ((a_|_ ->1 d) ^ (c_|_ ->1 d))) ^ (((b ->1 d) ^ (c ->1 d)) v ((b_|_ ->1 d) ^ (c_|_ ->1 d))))))
31	4, 30	2or 67	. . . 4 ((f ->1 d) v ((e ->1 d) ^ (((e ^ f) v ((e ->1 d) ^ (f ->1 d))) v (((e ^ g) v ((e ->1 d) ^ (g ->1 d))) ^ ((f ^ g) v ((f ->1 d) ^ (g ->1 d))))))) = ((a_|_ ->1 d) v ((b_|_ ->1 d) ^ ((((a ->1 d) ^ (b ->1 d)) v ((a_|_ ->1 d) ^ (b_|_ ->1 d))) v ((((a ->1 d) ^ (c ->1 d)) v ((a_|_ ->1 d) ^ (c_|_ ->1 d))) ^ (((b ->1 d) ^ (c ->1 d)) v ((b_|_ ->1 d) ^ (c_|_ ->1 d)))))))
32	1, 31	2an 72	. . 3 (f ^ ((f ->1 d) v ((e ->1 d) ^ (((e ^ f) v ((e ->1 d) ^ (f ->1 d))) v (((e ^ g) v ((e ->1 d) ^ (g ->1 d))) ^ ((f ^ g) v ((f ->1 d) ^ (g ->1 d)))))))) = ((a ->1 d) ^ ((a_|_ ->1 d) v ((b_|_ ->1 d) ^ ((((a ->1 d) ^ (b ->1 d)) v ((a_|_ ->1 d) ^ (b_|_ ->1 d))) v ((((a ->1 d) ^ (c ->1 d)) v ((a_|_ ->1 d) ^ (c_|_ ->1 d))) ^ (((b ->1 d) ^ (c ->1 d)) v ((b_|_ ->1 d) ^ (c_|_ ->1 d))))))))
33	32	ax-r1 34	. 2 ((a ->1 d) ^ ((a_|_ ->1 d) v ((b_|_ ->1 d) ^ ((((a ->1 d) ^ (b ->1 d)) v ((a_|_ ->1 d) ^ (b_|_ ->1 d))) v ((((a ->1 d) ^ (c ->1 d)) v ((a_|_ ->1 d) ^ (c_|_ ->1 d))) ^ (((b ->1 d) ^ (c ->1 d)) v ((b_|_ ->1 d) ^ (c_|_ ->1 d)))))))) = (f ^ ((f ->1 d) v ((e ->1 d) ^ (((e ^ f) v ((e ->1 d) ^ (f ->1 d))) v (((e ^ g) v ((e ->1 d) ^ (g ->1 d))) ^ ((f ^ g) v ((f ->1 d) ^ (g ->1 d))))))))
34		oa4gto4u.1	. . 3 ((e ->1 d) ^ (((e ^ f) v ((e ->1 d) ^ (f ->1 d))) v (((e ^ g) v ((e ->1 d) ^ (g ->1 d))) ^ ((f ^ g) v ((f ->1 d) ^ (g ->1 d)))))) =< (f ->1 d)
35	34	oaur 910	. 2 (f ^ ((f ->1 d) v ((e ->1 d) ^ (((e